In the previous sections, you studied the aspects of reasoning and proof in a problem-solving context; looked at ways to encourage students to develop their ability to make conjectures and to prove and extend their ideas; and considered the aspects of reasoning and proof in a middle-grades classroom.

Now that you are familiar with the standard and some of its aspects, you will have a chance to see how one teacher applies it. Middle school teacher Hilory Paster wants to evaluate how well students in her fifth-grade math class understand fractions.

How does reasoning and proof fit into this picture? First, consider a problem similar to the one Ms. Paster sets for her students.

As you saw in Part A of Session 2: Communication, Ms. Paster introduced her class to the Fraction Tracks game. In addition to this linear model for working with fractions, some students might use hands-on materials to help them with the game. There are many manipulative materials for fractions. We will show the Fraction Factory Pieces here, which are based on an area model for fractions. Students can use the pieces to help them decide which moves would add to the number they draw, or they can use the pieces to "prove" that their choices are correct.